1. **Problem Statement:** Find the area between $z=0$ and given $z$-scores using the z-table (standard normal distribution). 2. **Using the z-table:** The z-table gives the area from the mean (z=0) to a positive z-value. For negative z-values, use symmetry: area from 0 to $-z$ equals area from 0 to $z$. 3. **Part I: Areas between $z=0$ and given z-values** A. $z=-1$ - Area from 0 to $-1$ equals area from 0 to 1. - From z-table, area(0 to 1) = 0.3413. B. $z=2.56$ - From z-table, area(0 to 2.56) = 0.4948. C. $z=-1.28$ - Area from 0 to $-1.28$ equals area from 0 to 1.28. - From z-table, area(0 to 1.28) = 0.3997. 4. **Part II: Convert raw scores to z-scores and find area between them** Formula for z-score: $$z = \frac{X - \mu}{\sigma}$$ where $X$ is the raw score, $\mu=48$ is the mean, and $\sigma=6$ is the standard deviation. A. Scores 54 and 42 - $z_{54} = \frac{54-48}{6} = 1$ - $z_{42} = \frac{42-48}{6} = -1$ - Area between $z=-1$ and $z=1$ is twice area from 0 to 1: $$2 \times 0.3413 = 0.6826$$ B. Scores 52 and 60 - $z_{52} = \frac{52-48}{6} = \frac{4}{6} = 0.67$ - $z_{60} = \frac{60-48}{6} = 2$ - Area from 0 to 0.67 = 0.2514 (from z-table) - Area from 0 to 2 = 0.4772 - Area between 0.67 and 2 = $0.4772 - 0.2514 = 0.2258$ - Since 52 < 60 and both positive z, total area between 0.67 and 2 is 0.2258 C. Scores 55 and 60 - $z_{55} = \frac{55-48}{6} = 1.17$ - $z_{60} = 2$ (from above) - Area(0 to 1.17) = 0.3790 - Area(0 to 2) = 0.4772 - Area between 1.17 and 2 = $0.4772 - 0.3790 = 0.0982$ D. Scores 48 and 42 - $z_{48} = 0$ - $z_{42} = -1$ - Area between 0 and -1 equals area(0 to 1) = 0.3413 E. Scores 54 and 38 - $z_{54} = 1$ - $z_{38} = \frac{38-48}{6} = -1.67$ - Area(0 to 1) = 0.3413 - Area(0 to 1.67) = 0.4525 - Total area between -1.67 and 1 is sum of areas from 0 to 1 and 0 to 1.67: $$0.3413 + 0.4525 = 0.7938$$ **Final answers:** I. A. 0.3413 B. 0.4948 C. 0.3997 II. A. 0.6826 B. 0.2258 C. 0.0982 D. 0.3413 E. 0.7938 These areas represent probabilities or proportions of the standard normal distribution between the given z-values or raw scores.